| L(s) = 1 | − 7-s − 5·11-s − 6·13-s + 4·17-s + 6·19-s + 3·23-s − 3·29-s + 2·31-s + 7·37-s + 4·41-s − 7·43-s − 2·47-s + 49-s + 10·53-s − 14·59-s − 4·61-s + 3·67-s + 13·71-s − 16·73-s + 5·77-s + 79-s − 10·83-s − 10·89-s + 6·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.50·11-s − 1.66·13-s + 0.970·17-s + 1.37·19-s + 0.625·23-s − 0.557·29-s + 0.359·31-s + 1.15·37-s + 0.624·41-s − 1.06·43-s − 0.291·47-s + 1/7·49-s + 1.37·53-s − 1.82·59-s − 0.512·61-s + 0.366·67-s + 1.54·71-s − 1.87·73-s + 0.569·77-s + 0.112·79-s − 1.09·83-s − 1.05·89-s + 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.96973956336672, −13.45761156358703, −12.96907387182545, −12.56252974650438, −12.11475356093958, −11.57780793339418, −11.10471583477590, −10.35875016017742, −10.04800480961588, −9.644607798062390, −9.221014376775159, −8.434155130367958, −7.823378508657649, −7.417532348240788, −7.267365963132125, −6.364665681742161, −5.721499164563916, −5.215691878329378, −4.952071121627380, −4.231262383180677, −3.368192547492229, −2.807475507915697, −2.596478836322077, −1.621730918196936, −0.7562026413018636, 0,
0.7562026413018636, 1.621730918196936, 2.596478836322077, 2.807475507915697, 3.368192547492229, 4.231262383180677, 4.952071121627380, 5.215691878329378, 5.721499164563916, 6.364665681742161, 7.267365963132125, 7.417532348240788, 7.823378508657649, 8.434155130367958, 9.221014376775159, 9.644607798062390, 10.04800480961588, 10.35875016017742, 11.10471583477590, 11.57780793339418, 12.11475356093958, 12.56252974650438, 12.96907387182545, 13.45761156358703, 13.96973956336672
